Abstract
In this paper we investigate the solutions and the Hyers-Ulam stability of the μ-Jensen functional equation
a variant of the μ-Jensen functional equation
and the μ-quadratic functional equation
where S is a semigroup, σ is a morphism of S and μ: \(S\longrightarrow \mathbb {C}\) is a multiplicative function such that μ(xσ(x)) = 1 for all x ∈ S.
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References
Aczél, J., Dhombres, J.: Functional Equations in Several Variables. With Applications to Mathematics, Information Theory and to the Natural and Social Sciences. Encyclopedia of Mathematics and Its Applications, vol. 31. Cambridge University Press, Cambridge (1989)
Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950)
Bouikhalene, B., Elqorachi, E., Rassias, Th.M.: On the generalized Hyers-Ulam stability of the quadratic functional equation with a general involution. Nonlinear Funct. Anal. Appl. 12(2), 247–262 (2007)
Bouikhalene, B., Elqorachi, E., Redouani, A.: Hyers-Ulam stability of the generalierd quadratic functional equation in Amenables semigroups. J. Inequal. Pure Appl. Math. (JIPAM) 8(2), 18 pp. (2007). Article 56
Cholewa, P.W.: Remarks on the stability of functional equations. Aequationes Math. 27, 76–86 (1984)
Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg 62, 59–64 (1992)
Dilian, Y.: Contributions to the theory of functional equations. PhD Thesis, University of Waterloo, Waterloo, Ontario, Canada (2006)
Elqorachi, E., Rassias M.Th.: Generalized Hyers-Ulam stability of trigonometric functional equations. Mathematics 6(5), 11 pp. (2018)
Elqorachi, E., Manar, Y., Rassias, Th.M.: Hyers-Ulam stability of Wilson’s functional equation. In: Pardalos, P.M., Rassias, Th.M. (eds.) Contributions in Mathematics and Engineering: Honor of Constantin Carathéodory. Springer, New York (2016)
Elqorachi, E., Redouani, A.: Solutions and stability of a variant of Wilson’s functional equation. Proyecciones J. Math. 37(2), 317–344 (2018)
Faiziev, V.A., Sahoo. P.K.: On the stability of Jensen’s functional equation on groups. Proc. Indian Acad. Sci. (Math. Sci.) 117, 31–48 (2007)
Forti, G.L.: Hyers-Ulam stability of functional equations in several variables. Aequationes Math. 50, 143–190 (1995)
Gadja, Z.: On stability of additive mapping. Int. J. Math. Math. Sci. 14, 431–434 (1991)
Gǎvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)
Greenleaf, F.P.: Invariant Means on topological Groups and their Applications. Van Nostrand, New York (1969)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U. S. A. 27, 222–224 (1941)
Hyers, D.H., Isac, G.I., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkh\(\ddot {a}\)user, Basel (1998)
Jung, S-M.: Hyers-Ulam-Rassias stability of Jensen’s equation and its application. Proc. Amer. Math. Soc. 126, 3137–3134 (1998)
Jung, S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48. Springer, New York (2011)
Kim, J.H., The stability of d’Alembert and Jensen type functional equations. J. Math. Anal. Appl. 325, 237–248 (2007)
Ng, C.T.: Jensen’s functional equation on groups, III. Aequationes Math. 62(1–2), 143–159 (2001)
Rassias, Th.M.: On the stability of linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)
Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–120 (1982)
Rassias, M.Th.: Solution of a functional equation problem of Steven Butler. Octogon Math. Mag. 12, 152–153 (2004)
Stetkær, H.: Functional equations on abelian groups with involution. Aequationes Math. 54, 144–172 (1997)
Stetkær, H.: Functional Equations on Groups. World Scientific Publishing, New Jersey (2013)
Stetkær, H.: D’Alember’s functional equations on groups. Banach Cent. Publ. 99, 173–191 (2013)
Stetkær, H.: A variant of d’Alembert’s functional equation. Aequationes Math. 89(3), 657–662 (2015)
Stetkær, H.: A note on Wilson’s functional equation. Aequationes Math. 91(5), 945–947 (2017)
Szekelyhidi, L.: Note on a stability theorem. Can. Math. Bull. 25(4), 500–501 (1982)
Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publications, New york (1961). Problems in Modern Mathematics (Wiley, New York, 1964)
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Belfakih, K., Elqorachi, E., Rassias, T.M. (2019). Solutions and Stability of Some Functional Equations on Semigroups. In: Brzdęk, J., Popa, D., Rassias, T. (eds) Ulam Type Stability . Springer, Cham. https://doi.org/10.1007/978-3-030-28972-0_9
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